ABSTRACT


Toeplitz matrices enjoy the dual virtues of beauty and ubiquity. We begin this talk by surveying some of the interesting spectral properties of such matrices, emphasizing the distinctions between infinitedimensional Toeplitz matrices and the largedimensional limit of the corresponding finite matrices. These basic results utilize the algebraic Toeplitz structure, but in many applications, one is forced to spoil this structure with some perturbations (e.g., by imposing boundary conditions upon a finite difference discretization of an initialboundary value problem). How do such perturbations affect the eigenvalues? This talk will address this question for "localized" perturbations, i.e., perturbations that are restricted to a single entry, or a block of entries whose size remains fixed as the matrix dimension grows. One can show, for a broad class of matrices, that sufficiently small perturbations fail to alter the spectrum, though the spectrum is exponentially sensitive to other kinds of perturbations. For larger real singleentry perturbations, one observes the perturbed eigenvalues trace out curves in the complex plane. We'll show a number of illustrations of this phenomenon for tridiagonal Toeplitz matrices, which arise in many applications. This talk describes collaborative work with Albrecht Boettcher, Marko Lindner, and Viatcheslav Sokolov of TU Chemnitz. Return to Applied Math Colloquium home page 